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$V=\mathbb C^n$ is a $\mathbb CS_n$-module, where $S_n$ is the symmetric group of degree $n$, via the representation sending a permutation to the corresponding permutation matrix. The tensor power $V^...

Embed the hyperoctahedral group $H_n$ into the symmetric group $S_{2n}$ as the centralizer of the involution $(1, 2) (3, 4) \cdots (2n-1, 2n)$ (cycle notation). Label representations of $S_{2n}$ by ...

Background For $\lambda$ any partition and $n$ a positive integer, write $\lambda[n]$ for the sequence $(n - |\lambda|, \lambda_1, \lambda_2, \ldots, \lambda_r)$. For $n$ large enough, this is a ...

The fundamental theorem of symmetric polynomials tells us that the ring $\mathbb{Z}[x_1,\ldots,x_n]^{S_n}$ of symmetric polynomials in $n$ variables is generated (without relations) by the elementary ...

I came across the following cute fact about partitions: \begin{align} & |\{\lambda \vdash n \text{ with an even number of even parts}\}| \\[8pt] & {} - |\{ \lambda \vdash n \text{ with an odd ...